Your question reads at first like an advertising... That said:
1) Which party hold the responsibility to support/resolve the bugs of Linux kernel when EOS is in production?
If this is a vanilla kernel, you can submit bug report to the LKML for sure. NONE has a responsibility to do fix your bugs. Someone may help. This is a gift, not an obligation. If this is a patched kernel with proprietary LKMs it will likely be dismissed and ignored politely or with fracas by the kernel maintainers.
The commercial distributor may offer most likely a support contract and may accept such responsibility otherwise.
Short of this, none but YOU has the responsibility to support free and open source you use. Anything else is a courtesy and a gift from the project authors, maintainers and any willing good soul.
2) Does Arista (vendor) suppose to distribute EOS (commercial product) with source code or as closed source?
A commercial distributor of a Linux kernel and distro that contains GPL-licensed code has obligations to provide the corresponding source code to
the parties they redistribute binaries to. I am sure that this vendor does this alright like the vast majority of Linux distributors do and like the vast majority of device vendors that bundle a kernel do as well. If they sell this distribution as they are entitled to do so, their redistribution of source code can be limited to their paying customers that received the binaries. The same applies if they sell devices that contain a Linux kernel. They have no other general obligation. And certainly no obligation beyond the recipients of the binaries, such as making source code available to the general public.
3) What is the significance to mention MODULE_LICENSE("GPL") in every Linux kernel module?
This tag informs the kernel (and in particular through the use of the
insmod and other kernel module loading utilities) that this module declares itself as GPL-licensed and in return the kernel grants access to this module to kernel symbols exported as GPL-only and to kernel internals. As such it could interpreted either as weak indication of licensing and not much more that than. And as an indication that this module may depend on and require access to GPL-symbols exported by the kernel.